Methods Ecol Evol. 2021;12:1093–1102.  |  1093wileyonlinelibrary.com/journal/mee3

Received: 13 December 2020  |  Accepted: 8 February 2021

DOI: 10.1111/2041-210X.13583

A P P L I C A T I O N

rasterdiv— An Information Theory tailored R package for measuring ecosystem heterogeneity from space: To the origin and back

Duccio Rocchini1,2  | Elisa Thouverai1 | Matteo Marcantonio3  | Martina Iannacito4 | Daniele Da Re5  | Michele Torresani6 | Giovanni Bacaro7  | Manuele Bazzichetto8  | Alessandra Bernardi9  | Giles M. Foody10  | Reinhard Furrer11,12  | David Kleijn13  | Stefano Larsen14,15  | Jonathan Lenoir16  | Marco Malavasi2  | Elisa Marchetto1 | Filippo Messori1 | Alessandro Montaghi17 | Vítězslav Moudrý2  | Babak Naimi18  | Carlo Ricotta19  | Micol Rossini20  | Francesco Santi1 | Maria J. Santos21  | Michael E. Schaepman22  | Fabian D. Schneider23  | Leila Schuh11 | Sonia Silvestri1  | Petra Ŝímová2  | Andrew K. Skidmore24,25  | Clara Tattoni26  | Enrico Tordoni7  | Saverio Vicario27 | Piero Zannini1  | Martin Wegmann28

1BIOME Lab, Department of Biological, Geological and Environmental Sciences, Alma Mater Studiorum University of Bologna, Bologna, Italy; 2Department of Spatial Sciences, Faculty of Environmental Sciences, Czech University of Life Sciences Prague, Praha - Suchdol, Czech Republic; 3Department of Pathology, Microbiology, and Immunology, School of Veterinary Medicine, University of California, Davis, CA, USA; 4Inria Bordeaux - Sud- Ouest, Talence, France; 5Georges Lemaître Center for Earth and Climate Research, Earth and Life Institute, UCLouvain, Louvain- la- Neuve, Belgium; 6Faculty of Science and Technology, Free University of Bolzano/Bozen, Piazza Universitá/Universitätsplatz 1, Bolzano, Italy; 7Department of Life Sciences, University of Trieste, Trieste, Italy; 8EcoBio (Ecosystèmes, Biodiversité, Évolution) - UMR 6553, Université de Rennes, CNRS, Rennes, France; 9Department of Mathematics, University of Trento, Povo, Italy; 10School of Geography, University of Nottingham, Nottingham, UK; 11Department of Mathematics, University of Zurich, Zurich, Switzerland; 12Department of Computational Science, University of Zurich, Zurich, Switzerland; 13Plant Ecology and Nature Conservation Group, Wageningen University, Wageningen, The Netherlands; 14Unit of Computational Biology, Research and Innovation Center, Fondazione Edmund Mach, San Michele all'Adige, Italy; 15Department of Civil, Environmental and Mechanical Engineering, University of Trento, Trento, Italy; 16UR “Ecologie et Dynamique des Systèmes Anthropisés” (EDYSAN, UMR 7058 CNRS- UPJV), Université de Picardie Jules Verne, Amiens, France; 17DAGRI Department of Agriculture, Food, Environment and Forestry, University of Florence, Firenze, Italy; 18Department of Geosciences and Geography, University of Helsinki, Helsinki, Finland; 19Department of Environmental Biology, University of Rome “La Sapienza'”, Rome, Italy; 20Remote Sensing of Environmental Dynamics Laboratory, DISAT, Universitá degli Studi Milano- Bicocca, Milano, Italy; 21Department of Geography, Earth System Science, University of Zurich, Zurich, Switzerland; 22Department of Geography, Remote Sensing Laboratories, University of Zurich, Zurich, Switzerland; 23Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA, USA; 24Faculty of Geo- Information Science and Earth Observation (ITC), University of Twente, Enschede, The Netherlands; 25Department of Environmental Science, Macquarie University, Sydney, NSW, Australia; 26Department of Agriculture, Food, Environment and Forestry (DAGRI), University of Florence, Firenze, Italy; 27CNR- IIA C/O Physics Department “M. Merlin” University of Bari, Bari, Italy and 28Department of Remote Sensing, University of Wuerzburg, Würzburg, Germany

This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.© 2021 The Authors. Methods in Ecology and Evolution published by John Wiley & Sons Ltd on behalf of British Ecological Society

CorrespondenceDuccio RocchiniEmail: duccio.rocchini@unibo.it

Funding informationH2020 Project SHOWCASE, Grant/Award Number: 862480; H2020 COST Action, Grant/Award Number: CA17134; Jet Propulsion Laboratory; California Institute of Technology; National Aeronautics

Abstract1. Ecosystem heterogeneity has been widely recognized as a key ecological indica-

tor of several ecological functions, diversity patterns and change, metapopulation dynamics, population connectivity or gene flow.

2. In this paper, we present a new R package— rasterdiv— to calculate heterogeneity indices based on remotely sensed data. We also provide an ecological application

1094  |    Methods in Ecology and Evolu on ROCCHINI et al.

1  | INTRODUC TION

Ecosystem heterogeneity is related to a number of ecological pro-cesses and functions such as species diversity patterns and change (Rocchini et al., 2018), metapopulation dynamics (Fahrig, 2007), population connectivity (Malanson & Cramer, 1999) or gene flow (Lozier et al., 2013). Heterogeneity has been defined in various ways in the scientific literature: (a) as the variation in space and time of qualitative and quantitative descriptors of an environmental variable of interest (Li & Reynolds, 1995); (b) as the horizontal component of habitat variation (August, 1983; Grelle, 2003); (c) as the spatially structured variability of the habitat (Ettema & Wardle, 2002); or (d) as within- habitat variability (Heaney, 2001; Hortal et al., 2009). In this paper, it will be considered as an umbrella concept representing the degree of non- uniformity in land cover, vegetation and physi-cal factors (topography, soil, topoclimate and microclimate; Stein et al., 2014).

Landscape heterogeneity across different spatial extents and over different temporal periods can be inferred by applying algo-rithms based on remote sensing and spatial ecology (Schimel & Schneider, 2019; Skidmore et al., 2011). Remotely sensed spectral heterogeneity measures of a landscape represent a valid alterna-tive to categorical land cover maps, which, especially in the case of non- homogeneous and complex landscape configurations (e.g. mosaic of crops and semi- natural forests), might suffer from over-simplification when investigated through land cover classes (Da Re et al., 2019; Rocchini et al., 2019). Heterogeneous landscapes should present higher spectral heterogeneity values compared to more ho-mogeneous landscapes within the same spatial extent (Rocchini & Ricotta, 2007). It follows that remotely sensed spectral heterogeneity can be profitably used to measure landscape heterogeneity in space and time to convey information on ecosystem processes and func-tioning (Schneider et al., 2017).

From this point of view, the development of Free and Open- Source algorithms to measure and monitor (i.e. repeated measures over time) landscape or ecosystem heterogeneity from space would allow robust, reproducible and standardized estimates of ecosystem functioning and services (Rocchini & Neteler, 2012). Furthermore, their intrinsic transparency, community- vetoing options, sharing and rapid availability are also valuable additions and reasons to move towards open- source options. Considering the different open- source software options, the R software

(R Core Team, 2020) is among the most widely used languages for statistics and modelling and different packages have already been devoted to remote sensing data processing for: (a) raster data man-agement (raster package; Hijmans & van Etten, 2020); (b) remote sensing data analysis (RStoolbox package; Leutner et al., 2019); (c) spectral species diversity mapping (biodivMapR package; Féret & Boissieu, 2020); (d) Sparse Generalized Dissimilarity Modelling based on remote sensing data (sgdm package; Leitão et al., 2017); (e) entropy- based local spatial association (ELSA package; Naimi et al., 2019); or (f) landscape metrics calculation (landscape-metrics package; Hesselbarth et al., 2019), to name just a few. Readers can also refer to https://cran.r- proje ct.org/web/views/ Spati al.html for the CRAN Task View on analysis of spatial data.

Nonetheless, no r package currently provides a flow of func-tions grounded in Information Theory and generalized entropy, incorporating abundance information for each informative value but also the relative numerical distance among said values. In this paper, we introduce the new rasterdiv r package, now available under the Comprehensive R Archive Network (CRAN, https://CRAN.R- proje ct.org/packa ge=raste rdiv), which provides such a set of functions' throughput workflow. The aim of this man-uscript is to briefly introduce the theory under the rasterdiv package and to provide an ecological example demonstrating its ability to measure several aspects of landscape or ecosystem heterogeneity.

2  | BRIEF DESCRIPTION OF THE THEORETIC AL FR AME WORK ON INFORMATION- THEORY- BA SED METRIC S

2.1 | Shannon entropy

Shannon's (1948) theory, profoundly rooted in Eduard Boltzmann's (1872) studies, is a solid basis for calculating landscape heteroge-neity and it has been widely used in several ecological applications (refer to Vranken et al., 2014 for a review).

Given a sample area with N pixel values and pi relative abun-dances for every i ∈ {1, …, N} the Shannon index is calculated as:

(1)H = −

N∑i=1

pi lnpi,

and Space Administration, Grant/Award Number: 80NM0018D0004; University of Zurich Research Priority Program on Global Change and Biodiversity (URPP GCB); Friuli Venezia Giulia Region, Swiss National Science Foundation, Grant/Award Number: SNSF– 175529; Czech University of Life Sciences Prague

Handling Editor: Sarah Goslee

at the landscape scale and demonstrate its power in revealing potentially hidden heterogeneity patterns.

3. The rasterdiv package allows calculating multiple indices, robustly rooted in Information Theory, and based on reproducible open- source algorithms.

K E Y W O R D S

biodiversity, ecological informatics, modelling, remote sensing, satellite imagery

     |  1095Methods in Ecology and Evolu onROCCHINI et al.

When applying such indices to remotely sensed data, the image is divided into small chunks of the whole image, commonly defined as ‘ker-nels’, ‘windows’ or ‘moving windows’ (see Figure 1 in Box 1). These terms will be used throughout this manuscript to relay the local scale of analysis.

In the Shannon index, the relative abundance of pixel values (e.g. reflectance values, vegetation indices) is considered. The higher the richness and turnover, the higher will be the equitability of values and thus the Shannon index.

BOX 1 Description of the moving window approach

Given a raster layer, ecologists usually split it into small chunks, called moving windows. To estimate the value of the previously presented indices, we first select l an odd number, which will correspond to the length of the squared window. With this choice, we have a central entry in the window, that is, the

(l+ 1

2,l+ 1

2

) entry, which will be placed as a mask, in position (1, 1) over our raster. The

index is therefore computed only with the values which the window covers. Notice that with this choice we will have some missing values, which will not contribute to the index computation. The obtained index value is stored in position (1, 1) in the output raster. The following step is moving the window so that its central entry is over the entry (1, 2) of the raster. The index value computed is stored in the corresponding position of the output raster, that is, in (1, 2). We proceed in this way until the last entry of the output raster is filled.

This technique, visually presented in Figure 1, is extremely popular in ecology since it is quite simple, robust and powerful to simu-late a biological boundary. Note that this technique of computing index has a suitable algorithmic structure.

F I G U R E 1   The moving window technique for the computation of diversity indices in the rasterdiv package, redrawn from Rocchini et al. (2013)

Algorithm 1 Index computation with moving window

1096  |    Methods in Ecology and Evolu on ROCCHINI et al.

2.2 | Rényi generalized entropy

Any point descriptor of information heterogeneity, like the previ-ously cited Shannon's H, is not able to describe the whole potential spectrum of heterogeneity as it usually measures only one part or component of heterogeneity (e.g. richness, evenness, nestedness, etc.). Hence, no single measure can be used to represent such a wide spectrum (Gorelick, 2011; Nakamura et al., 2020).

Rényi (1970) proposed a method to generalize entropy mea-surements in just one formula, changing one parameter, called � in its original formulation. Given a sample area with N pixel values and pi relative abundances for every i ∈ {1, …, N}, the Rényi en-tropy index is:

Changing the parameter � will lead to different indices starting from the same formula (Hill, 1973). As an example, when α = 0, H0 = ln(N) where N = richness, namely the maximum possible Shannon index (Hmax). In practice, with � = 0, all the spectral values equally contrib-ute to the index, without making use of their relative abundance. For � → 1, the Rényi entropy index will equal Shannon H, according to the l'Hôpital's rule (a mathematical proof is provided in Supporting Information Appendix 1), while for α = 2 the Rényi entropy index will equal the ln(1/D) where D is the Simpson's dominance (Simpson, 1949). The theoretical curve relating the Rényi entropy index and α is a nega-tive exponential, that is, it decays until flattening for higher values of α, where the weight of the most abundant spectral values is higher with small differences among the attained heterogeneity maps (Ricotta et al., 2003). Besides the Rényi's generalized entropy, the rasterdiv package includes generalized metrics as the Hill's (1973) numbers.

2.3 | Rao's Q heterogeneity index

The previously described metrics have no dimension. In other words, they do not consider the relative difference among pixel val-ues but just the presence of a different class. For example, having A = (1, 2, 3, 4, 5, 6, 7, 8, 9) and B = (1, 102, 103, 104, 105, 106, 107, 108, 109) as two theoretical arrays containing values that are different from each other, the Shannon index will always be maximum, that is, H = ln (9) = 2.197225 despite the relative numerical distance between pairs of values.

In remote sensing, this is a critical point since contiguous zones of a satellite image might have similar (but not strictly equal) reflec-tance values. For instance, the variability of a homogeneous sur-face like a woodland patch or a water area, would be overestimated within the landscape matrix if spectral distances among values are not considered in the calculation.

The Rao's Quadratic heterogeneity measure (hereafter Rao's Q; Rao, 1982) can be applied to overcome this issue, considering both relative abundances and spectral distances among pixel values in the calculation.

Given the values of different pixels i and j, the Rao's Q considers their pairwaise distance dij as:

Accordingly, an array with different but spectrally near values will convey a high Shannon's H but a low Rao's Q. Conversely, an array with different and distant spectral values will convey both a high Shannon's H and a high Rao's Q.

Given a 3 × 3 pixels matrix M:

where � is the reflectance value for every pixel in a single 8- bit band (256 possible values), a pairwise distance matrix Md is derived for all pixel values:

Then, according to Equation (3), Rao's Q is obtained as the sum of every pairwise distance multiplied by the relative abundances of all the pairs of pixels in the analysed image d × (1∕N2). Strictly speaking, Rao's Q can be defined as the expected difference in reflectance values between two pixels drawn randomly with replacement from the evaluated set of pixels.

It is possible to construct the distance matrix for several dimen-sions (layers) to consider multiple bands at a time and calculate Rao's Q in a multidimensional (multi- layer) system.

To illustrate the benefit of the functions provided in the rasterdiv package, we apply it on an ecological study case in the following section.

3  | APPLIC ATION: THE ECOSYSTEM HETEROGENEIT Y OF THE ÖTZI ARE A

To show the capabilities of the rasterdiv package, we decided to focus on one of the geologically and biologically most diverse mountain regions worldwide: the Similaun and Ortles glaciers in Italy. This region is not only important for its rich geobiodiver-sity but also for its fascinating archaeological history, also due to an incredible anthropological discovery of the early nineties: the famous Ötzi Tyrolean iceman (Keller et al., 2012). The study area we are focusing on for applying the rasterdiv package is

(2)H�=

1

1 − �× ln

N∑i=1

p�i.

(3)Q =

N∑i=1

N∑j=1

dij × pi × pj.

M =

⎛⎜⎜⎜⎜⎝

�1 �2 �3

�4 �5 �6

�7 �8 �9

⎞⎟⎟⎟⎟⎠,

(4)Md =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

d�1,�1

d�1,�2

d�1,�3

⋯ d�1,�N

d�2,�1

d�2,�2

d�2,�3

⋯ d�2,�N

d�3,�1

d�3,�2

d�3,�3

⋯ d�3,�N

⋮ ⋮ ⋮ ⋱ ⋮

d�N ,�1

d�N,�2

d�N ,�3

⋯ d�N ,�N

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

.

     |  1097Methods in Ecology and Evolu onROCCHINI et al.

included between the Ortles and the Similaun glaciers, in the Alps of northern Italy (Figure 2). Below, we provide a step- by- step tutorial which can be reproduced for any area and by every researcher worldwide. The only required input data are satellite images.

We used Copernicus Sentinel- 2 data at a spatial resolution of 10 m (Figure 2) acquired on May 9th 2020. Once the satel-lite image was downloaded from the Copernicus Open Access Hub (https://scihub.coper nicus.eu/), we computed NDVI and, for the purposes of this research, rescaled it to an 8- bit radio-metric resolution. Hence, NDVI values were used as the input information (values) to compute the ecosystem heterogene-ity indices described in the former section. More specifically, based on the NDVI raster grid used as input object, we ran a set of functions provided in the rasterdiv package and written in Box 2.

We applied the code written in Box 2, using a moving window of 9 × 9 pixels, to both (a) the Similaun glacier upper area, mainly covered by alpine conifer woodlands (dominated by Picea abies, Larix decidua, Abies alba) and rocks, and (b) to the valley bottom (Val Venosta), a human- dominated landscape devoted to agricultural areas and small urban villages.

Concerning the Similaun glacier area, the Shannon index showed medium to high values everywhere, including areas with almost ho-mogeneous rock cover (i.e. alpine habitat) as well as areas with ho-mogeneous tree cover (i.e. coniferous forest habitat) (Figure 3). This is due to the fact that Shannon's H does not take into account the distance among pixel values but only the relative abundance of each value within the moving window of 9 × 9 pixels. In this case, NDVI values showed subtle differences among each other, especially in homogeneous areas and even when rescaled at 8 bit (namely 256 possible integer values).

F I G U R E 2   The Ötzi area in the northern Italian Alps, used for building an ecological example of the application of the rasterdiv package starting from a Copernicus Sentinel- 2 image, represented by an RGB space in natural (red, green and blue) and false (near infrared, red and green) colours. Coordinates are in the UTM (WGS84, zone 32N) reference system

5,200,000

5,160,000

5,120,000

600,000 630,000 660,000 690,000 600,000 630,000 660,000 690,000

5,200,000

5,160,000

5,120,000

BOX 2 Code used under rasterdiv

1098  |    Methods in Ecology and Evolu on ROCCHINI et al.

Low heterogeneity values were found in areas with snow cover; in that case, the values of the neighbouring pixels are the same and they lead to a low Shannon's H for the focal pixel centered on the moving window. The saturation of high values of heterogeneity was apparent when considering Rényi's entropy at low alpha values (Equation 2). This is related to the aforementioned (see Section 2.2) negative exponential curve relating the value of this index with re-spect to alpha, shown in Ricotta et al. (2003). The result is a map with saturated values of heterogeneity. This effect is softening when increasing the alpha parameter. The rasterdiv package allows accounting for several indices at a time to avoid heterogeneity sat-uration effects, for instance by considering distance among pixel values besides relative abundance of each value. More specifically, running the Rao's Q function coded in rasterdiv allows to circumvent this issue of saturated values of heterogeneity (refer to the bottom line of Figure 3). In the Similaun glacier area, the homogeneous cover of spruce forests is better reflected by the Rao's Q index as it better contrasts against the geological heterogeneity of the upper alpine belt. Maximum Rao's Q values were found at the interface between water (i.e. alpine lakes) and the surrounding vegetation and rocks, representing an interesting ecotone area (see the upper right corner of the last panel in Figure 3).

The rasterdiv package, thanks to a combination of functions rooted in Information Theory, thus helps to reveal hidden spatial pat-terns of heterogeneity, and allows measuring ecosystem heteroge-neity related to both biotic and abiotic components. By doing so, the

rasterdiv package better discriminates among ecosystem features and functions constituting the landscape.

In the valley area (Val Venosta), this phenomenon was even more pronounced (Figure 4) due to the higher spatial heterogeneity of the landscape matrix under study with patches of small agricultural fields mixed with water bodies (e.g. the Adige river in the middle of the valley), resulting in very high values for both Rényi (α = 0) and Shannon indices. Low to medium values on the north- and south- facing slopes indicate grasslands and broadleaf forests. Again, the Rao's Q index helped to better discriminate among these areas and among different land uses. Indeed, by relying on the relative numer-ical distance among the 8- bit- NDVI pixel values, it can differenti-ate between areas with low to medium heterogeneity values (blue and light blue colours in Figure 4), that is, grasslands and broadleaf forests, and areas with medium to high heterogeneity values, that is, upper mountain rocks at the interface with the treeline and with alpine lakes, as well as riparian vegetation besides the Adige river in the lower part of the valley.

With this study case focusing on a very heterogeneous area in the Alps, we illustrated two main critical components of the ras-terdiv package: (a) how it allows users to measure multiple indices simultaneously and (b) how it can help detecting otherwise hidden patterns of heterogeneity in the landscape.

For the sake of clarity and to avoid a catalogue- like article, we did not present all the indices that can be calculated by the raster-div package. Instead, we decided to showcase a few, to illustrate to

F I G U R E 3   The first row of the figure represents the area under study, namely the Similaun glacier as a subset of the study area in Figure 2. NDVI is shown in the natural range but it was rescaled to 8- bit before heterogeneity computation. Then, different metrics were applied: the Rényi's entropy index, the Shannon's H (corresponding to the Rényi's entropy with α = 1) and the Rao's Q. Coordinates are in the UTM (WGS84, zone 32N) reference system

     |  1099Methods in Ecology and Evolu onROCCHINI et al.

ecologists the high potential of the rasterdiv package for applica-tions in landscape ecology, macroecology, biogeography and the analysis of spatiotemporal dynamics in general. We refer to the man-ual of the package (https://CRAN.R- proje ct.org/packa ge=raste rdiv) and its vignettes to find additional metrics and examples related to the package.

4  | DISCUSSION

In this paper, we provide an ecological overview of new r pack-age rasterdiv. The overarching rationale for proposing this new r package is to present a set of methods and ready- to- use func-tions for calculating landscape heterogeneity metrics from space (e.g. satellite images), as well as from airborne or ground- based devices, to monitor and analyse, among other things, bio-diversity change, habitat fragmentation and land use and cover changes.

The importance of computing continuous spectral heterogene-ity measures from satellite- borne or airborne sensors to better dis-criminate among the various components in the landscape has been highlighted in several studies (Doxa & Prastacos, 2020; Godinho et al., 2018; Karlson et al., 2015; Ribeiro et al., 2019). Nevertheless, caution is recommended when making use of continuous remotely sensed data, and the radiometry of pixel values should be carefully considered before applying such measures. For instance, relying on float (decimal) precision data such as the NDVI (which ranges from −1 to 1) may lead to a high neighbouring heterogeneity which could

actually be the effect of data binning rather than the effect of an eco-logical underlying pattern. In general, an 8- bit image (and therefore composed of 256 integer values/classes) is preferable when applying spectral heterogeneity measures. In this paper, we used an 8- bit NDVI layer rescaled from Copernicus data. However, one might even rely on a multispectral system reduced to a single 8- bit layer through by means of the first component of a principal component analysis or any multidimensionality reduction technique (Féret & Boissieu, 2020).

Most metrics based on Information Theory can accommodate only one layer at a time when relying on indices using abundance information only (in the shown example, the Rényi entropy index and Shannon's H). However, the rasterdiv package includes accounting for pixel values distances, such as the Rao's Q index, which can inte-grate multiple layers of ecological information such as multiple bands of satellite images or physical and biotic data (e.g. vegetation cover, soil pH, topography).

In general, remotely sensed data are simplifications of more com-plex systems depending on the radiometric and spectral properties of one or more images. From an ecological point of view, the spectral space of an image might be associated with the Hutchinson's hyper-volume which orders geometrically the variables shaping species' ecological niches (Blonder, 2018; Hutchinson, 1959). Hence, calcu-lating heterogeneity in such a space could provide important infor-mation about species niches variability or at least on the landscape variability shaping species distribution (Rocchini et al., 2018).

Future local and global changes are expected to impact eco-system heterogeneity. Since remotely sensed data nowadays allow us to rely on relatively long and standardized time series,

F I G U R E 4   The first row of the figure represents the area under study, namely the Val Venosta, a subset at a lower elevation with respect to the Similaun glacier of Figure 3, and thus with higher human impact. NDVI is shown in the natural range but it was rescaled to 8- bit before heterogeneity computation. Then, different metrics were applied: the Rényi's entropy, the Shannon's H (corresponding to the Rényi's entropy with α = 1) and the Rao's Q. Coordinates are in the UTM (WGS84, zone 32N) reference system

1100  |    Methods in Ecology and Evolu on ROCCHINI et al.

applying different measures of heterogeneity to multi- temporal stacks would enhance the power to estimate and potentially fore-cast ecosystem heterogeneity shifts in space and time. This will be an invaluable tool to allow targeted and efficient monitoring and planning practices. For instance, due to the unprecedented rate of climate change, the adaptation of species to climate change is a benchmark in ecology (Stein et al., 2014). The rasterdiv pack-age might also be particularly useful when aiming at calculating climate- related heterogeneity, which is likely to shape ecosystem heterogeneity patterns that species have adapted to. This could be directly done running the functions on remotely sensed climate data (Metz et al., 2014; Senner et al., 2018; Zellweger et al., 2019), which are expected to drive several ecological functions at differ-ent spatial scales.

5  | CONCLUSION

Measuring heterogeneity from space to understand ecologi-cal patterns and processes acting across the landscape and over different time periods is crucial to guide effective management practices, especially in the Anthropocene epoch, in which human intervention is leading to rapid environmental changes (Randin et al., 2020).

The proposed rasterdiv package is a powerful tool for mon-itoring spatial and temporal variation of ecosystems' properties (Rocchini et al., 2018), given the intrinsic relationship (sensu Laliberté et al., 2019) between the spatial variation of ecosystems and that of the spectral signal from pixel values (Rocchini et al., 2019). No single measure can provide a full description of all the different aspects of ecosystem heterogeneity. That is why the rasterdiv package offers multiple approaches to disentangle the complexity of ecosystem heterogeneity in space and time through calculations deeply rooted in Information Theory and based on reproducible Free and Open- Source algorithms.

ACKNOWLEDG EMENTSWe are grateful to the handling Editor and two anonymous review-ers who helped us improving a previous version of this manuscript with their precious suggestions. D.R. and D.K. were partially sup-ported by the H2020 Project SHOWCASE (Grant agreement No 862480). D.R. was also partially supported by the H2020 COST Action CA17134 ‘Optical synergies for spatiotemporal sensing of scalable ecophysiological traits (SENSECO)'. The research car-ried out at the Jet Propulsion Laboratory, California Institute of Technology, was under a contract with the National Aeronautics and Space Administration (80NM0018D0004). Government sponsorship is acknowledged. This work was supported by Friuli Venezia Giulia Region Operative Program, European Social Fund— 2014/2020 Program, Specific Action n. 53/16: Integrative profes-sional training courses within degree programs. RF was partially supported by SNSF- 175529. We thank the Czech University of Life Sciences Prague. Faculty of Environmental Sciences for supporting

their research group Spatial Sciences in Ecology and Environment. The contribution of R.F. M.J.S., M.E.S. and L.S. is supported by the University of Zurich Research Priority Program on Global Change and Biodiversity (URPP GCB).

AUTHORS' CONTRIBUTIONSD.R., M.I., E.T., M.Mar., D.D.R., G.B., E.M., C.T., S.V. and C.R. con-tributed to the development of the algorithms and the coding of the rasterdiv package. D.R., M.T., M.B., A.B., G.M.F., R.F., D.K., S.L., J.L., M.Mal., F.M., A.M., V.M., B.N., M.R., F.S., M.J.S., M.E.S., F.D.S., L.S., S.S., P.S., A.K.S., E.T., P.Z. and M.W. contributed to the conceptual development of the theoretical background of the rasterdiv pack-age. All authors contributed to the writing of the manuscript.

PEER RE VIE WThe peer review history for this article is available at https://publo ns.com/publo n/10.1111/2041- 210X.13583.

DATA AVAIL ABILIT Y S TATEMENTFree data for applying the proposed functions are available directly into the rasterdiv package (https://CRAN.R- proje ct.org/packa ge=raste rdiv). The Copernicus Sentinel- 2 image used in this paper can be freely downloaded from: https://scihub.coper nicus.eu/, image tile reference: \ S2A_MSIL2A_20200905T101031_N0214_R022_T32TPS_202009 05T130252.

ORCIDDuccio Rocchini https://orcid.org/0000-0003-0087-0594 Matteo Marcantonio https://orcid.org/0000-0003-3896-2355 Daniele Da Re https://orcid.org/0000-0002-3398-9295 Giovanni Bacaro https://orcid.org/0000-0003-0946-4496 Manuele Bazzichetto https://orcid.org/0000-0002-9874-5064 Alessandra Bernardi https://orcid.org/0000-0002-2970-2730 Giles M. Foody https://orcid.org/0000-0001-6464-3054 Reinhard Furrer https://orcid.org/0000-0002-6319-2332 David Kleijn https://orcid.org/0000-0003-2500-7164 Stefano Larsen https://orcid.org/0000-0002-6774-1407 Jonathan Lenoir https://orcid.org/0000-0003-0638-9582 Marco Malavasi https://orcid.org/0000-0002-9639-1784 Vítězslav Moudrý https://orcid.org/0000-0002-3194-451X Babak Naimi https://orcid.org/0000-0001-5431-2729 Carlo Ricotta https://orcid.org/0000-0003-0818-3959 Micol Rossini https://orcid.org/0000-0002-6052-3140 Maria J. Santos https://orcid.org/0000-0002-6558-7477 Michael E. Schaepman https://orcid.org/0000-0002-9627-9565 Fabian D. Schneider https://orcid.org/0000-0003-1791-2041 Sonia Silvestri https://orcid.org/0000-0003-3302-4335 Petra Ŝímová https://orcid.org/0000-0003-2480-1171 Andrew K. Skidmore https://orcid.org/0000-0002-7446-8429 Clara Tattoni https://orcid.org/0000-0003-1555-5669 Enrico Tordoni https://orcid.org/0000-0002-9722-6692 Piero Zannini https://orcid.org/0000-0003-2466-4402 Martin Wegmann https://orcid.org/0000-0003-0335-9601

     |  1101Methods in Ecology and Evolu onROCCHINI et al.

R E FE R E N C E SAugust, P. V. (1983). The role of habitat complexity and heterogeneity in

structuring tropical mammal communities. Ecology, 64, 1495– 1507.Blonder, B. (2018). Hypervolume concepts in niche- and trait- based ecol-

ogy. Ecography, 41, 1441– 1455. https://doi.org/10.1111/ecog.03187Boltzmann, L. E. (1872). Weitere studien über das waärmegleichgewicht

unter gasmolekälen. S K Akad Wiss Wein, 66, 275– 370.Da Re, D., De Clercq, E. M., Tordoni, E., Madder, M., Rousseau, R., &

Vanwambeke, S. O. (2019). Looking for ticks from space: Using re-motely sensed spectral diversity to assess Amblyomma and Hyalomma tick abundance. Remote Sensing, 11, 770.

Doxa, A., & Prastacos, P. (2020). Using Rao's quadratic entropy to de-fine environmental heterogeneity priority areas in the European Mediterranean biome. Biological Conservation, 241, 108366.

Ettema, C. H., & Wardle, D. A. (2002). Spatial soil ecology. Trends in Ecology & Evolution, 17, 177– 183.

Fahrig, L. (2007). Landscape heterogeneity and metapopulation dynam-ics. In J. Wu & R. Hobbs (Eds.), Key topics in landscape ecology (pp. 78– 91). Cambridge University Press.

Féret, J.- B., & de Boissieu, F. (2020). biodivMapR: An R package for α- and β- diversity mapping using remotely sensed images. Methods in Ecology and Evolution, 11, 64– 70.

Godinho, S., Guiomar, N., & Gil, A. (2018). Estimating tree canopy cover percentage in a mediterranean silvopastoral systems using Sentinel- 2A imagery and the stochastic gradient boosting algorithm. International Journal of Remote Sensing, 39, 4640– 4662.

Gorelick, R. (2011). Do we have a consistent terminology for species di-versity? The fallacy of true diversity. Oecologia, 167, 885– 888.

Grelle, C. E. V. (2003). Forest structure and vertical stratification of small mammals in a secondary Atlantic forest, southeastern Brazil. Studies on Neotropical Fauna and Environment, 38, 81– 85.

Heaney, L. R. (2001). Small mammal diversity along elevational gradients in the Philippines: An assessment of patterns and hypotheses. Global Ecology and Biogeography, 10, 15– 39.

Hesselbarth, M. H. K., Sciaini, M., With, K. A., Wiegand, K., & Nowosad, J. (2019). landscapemetrics: An open- source R tool to calculate land-scape metrics. Ecography, 42, 1648– 1657.

Hijmans, R. J., & van Etten, J. (2020). raster: Geographic analysis and modeling with raster data. R package version 3.0- 12. Retrieved from http://CRAN.R- proje ct.org/packa ge=raster

Hill, M. O. (1973). Diversity and evenness: A unifying notation and its con-sequences. Ecology, 54, 427– 431. https://doi.org/10.2307/1934352

Hortal, J., Triantis, K. A., Meiri, S., Thébault, E., & Sfenthourakis, S. (2009). Island species richness increases with habitat diversity. The American Naturalist, 174, E205– E217. https://doi.org/10.1086/645085

Hutchinson, G. (1959). Homage to Santa Rosalia or why are there so many kinds of animals? American Naturalist, 93, 145– 159.

Karlson, M., Ostwald, M., Reese, H., Sanou, J., Tankoano, B., & Mattsson, E. (2015). Mapping tree canopy cover and aboveground biomass in Sudano- Sahelian woodlands using Landsat 8 and Random Forest. Remote Sensing, 7, 10017– 10041.

Keller, A., Graefen, A., Ball, M., Matzas, M., Boisguerin, V., Maixner, F., Leidinger, P., Backes, C., Khairat, R., Forster, M., Stade, B., Franke, A., Mayer, J., Spangler, J., McLaughlin, S., Shah, M., Lee, C., Harkins, T. T., Sartori, A., … Zink, A. (2012). New insights into the Tyrolean Iceman's origin and phenotype as inferred by whole- genome sequencing. Nature Communications, 3, 698.

Laliberté, E., Schweiger, A. K., & Legendre, P. (2019). Partitioning plant spect-ral diversity into alpha and beta components. Ecology Letters, 23, 370– 380.

Leitão, P. J., Schwieder, M., & Senf, C. (2017). sgdm: An R package for performing Sparse Generalized Dissimilarity Modelling with tools for gdm. ISPRS International Journal of Geo- Information, 6, 23.

Leutner, B., Horning, N., Schwalb- Willmann, J., & Hijmans, R. J. (2019). RStoolbox: Tools for remote sensing data analysis. R package version 0.2.6. Retrieved from http://CRAN.R- proje ct.org/packa ge=RStoo lbox

Li, H., & Reynolds, J. (1995). On definition and quantification of hetero-geneity. Oikos, 73, 280– 284.

Lozier, J. D., Strange, J. P., & Koch, J. B. (2013). Landscape heterogeneity predicts gene flow in a widespread polymorphic bumble bee, Bombus bifarius (Hymenoptera: Apidae). Conservation Genetics, 14, 1099– 1110. https://doi.org/10.1007/s1059 2- 013- 0498- 3

Malanson, G. P., & Cramer, B. E. (1999). Landscape heterogeneity, con-nectivity, and critical landscapes for conservation. Diversity and Distributions, 5, 27– 39.

Metz, M., Rocchini, D., & Neteler, M. (2014). Surface temperatures at the continental scale: Tracking changes with remote sensing at unprece-dented detail. Remote Sensing, 6, 3822– 3840.

Naimi, B., Hamm, N. A. S., Groen, T. A., Skidmore, A. K., Toxopeus, A. G., & Alibakhshi, S. (2019). ELSA: Entropy- based local indicator of spatial association. Spatial Statistics, 29, 66– 88.

Nakamura, G., Goncalves, L. O., & Duarte, L. D. S. (2020). Revisiting the dimensionality of biological diversity. Ecography, 43(4), 539– 548. https://doi.org/10.1111/ecog.04574

R Core Team. (2020). R: A language and environment for statistical comput-ing. R Foundation for Statistical Computing.

Randin, C. F., Ashcroft, M., Bolliger, J., Cavender- Bares, J., Coops, N., Dullinger, S., Dirnböck, T., Eckert, S., Ellis, E., Giuliani, G., Guisan, A., Jetz, W., Joost, S., Karger, D., Lembrechts, J., Lenoir, J., Luoto, M., Morin, X., Price, B., … Payne, D. (2020). Monitoring biodiversity in the Anthropocene using remote sensing in species distribution models. Remote Sensing of Environment, 239, 111626.

Rao, C. R. (1982). Diversity and dissimilarity coefficients: A unified ap-proach. Theoretical Population Biology, 21, 24– 43.

Rényi, A. (1970). Probability Theory. North Holland Publishing Company.Ribeiro, I., Proenca, V., Serra, P., Palma, J., Domingo- Marimon, C., Pons,

X., & Domingos, T. (2019). Remotely sensed indicators and open- access biodiversity data to assess bird diversity patterns in mediter-ranean rural landscapes. Scientific Reports, 9, 1– 13.

Ricotta, C., Corona, P., Marchetti, M., Chirici, G., & Innamorati, S. (2003). LaDy: Software for assessing local landscape diversity profiles of raster land cover maps using geographic windows. Environmental Modelling & Software, 18, 373– 378.

Rocchini, D., Delucchi, L., Bacaro, G., Cavallini, P., Feilhauer, H., Foody, G. M., He, K. S., Nagendra, H., Porta, C., Ricotta, C., Schmidtlein, S., Spano, L. D., Wegmann, M., & Neteler, M. (2013). Calculating land-scape diversity with information- theory based indices: A GRASS GIS solution. Ecological Informatics, 17, 82– 93.

Rocchini, D., Luque, S., Pettorelli, N., Bastin, L., Doktor, D., Faedi, N., Feilhauer, H., Féret, J.- B., Foody, G. M., Gavish, Y., Godinho, S., Kunin, W. E., Lausch, A., Leitão, P. J., Marcantonio, M., Neteler, M., Ricotta, C., Schmidtlein, S., Vihervaara, P., … Nagendra, H. (2018). Measuring β- diversity by remote sensing: A challenge for biodiversity monitor-ing. Methods in Ecology and Evolution, 9, 1787– 1798.

Rocchini, D., Marcantonio, M., Da Re, D., Chirici, G., Galluzzi, M., Lenoir, J., Ricotta, C., Torresani, M., & Ziv, G. (2019). Time- lapsing biodiversity: An open source method for measuring diversity changes by remote sensing. Remote Sensing of Environment, 231, 111192.

Rocchini, D., & Neteler, M. (2012). Let the four freedoms paradigm apply to ecology. Trends in Ecology & Evolution, 27, 310– 311.

Rocchini, D., & Ricotta, C. (2007). Are landscapes as crisp as we may think? Ecological Modelling, 204, 535– 539. https://doi.org/10.1016/j.ecolm odel.2006.12.028

Schimel, D., & Schneider, F. D. (2019). Flux towers in the sky: Global ecology from space. New Phytologist, 224, 570– 584. https://doi.org/10.1111/nph.15934

Schneider, F. D., Morsdorf, F., Schmid, B., Petchey, O. L., Hueni, A., Schimel, D. S., & Schaepman, M. E. (2017). Mapping functional di-versity from remotely sensed morphological and physiological forest traits. Nature Communications, 8, 1441.

1102  |    Methods in Ecology and Evolu on ROCCHINI et al.

Senner, N. R., Stager, M., & Cheviron, Z. A. (2018). Spatial and temporal heterogeneity in climate change limits species' dispersal capabilities and adaptive potential. Ecography, 41, 1428– 1440.

Shannon, C. E. (1948). A mathematical theory of communication. Bell System Technical Journal, 27(379– 423), 623– 656. https://doi.org/10.1002/j.1538- 7305.1948.tb009 17.x

Simpson, E. H. (1949). Measurement of diversity. Nature, 163, 688.Skidmore, A. K., Franklin, J., Dawsonc, T. P., & Pilesjo, P. (2011). Geospatial

tools address emerging issues in spatial ecology: A review and com-mentary on the Special Issue. International Journal of Geographical Information Science, 25, 337– 365.

Stein, A., Gerstner, K., & Kreft, H. (2014). Environmental heterogeneity as a universal driver of species richness across taxa, biomes and spa-tial scales. Ecology Letters, 17(7), 866– 880.

Vranken, I., Baudry, J., Aubinet, M., Visser, M., & Bogaert, J. (2014). A review on the use of entropy in landscape ecology: Heterogeneity, unpredictability, scale dependence and their links with thermody-namics. Landscape Ecology, 30, 51– 65.

Zellweger, F., De Frenne, P., Lenoir, J., Rocchini, D., & Coomes, D. (2019). Advances in microclimate ecology arising from remote sensing. Trends in Ecology & Evolution, 34, 327– 341.

SUPPORTING INFORMATIONAdditional supporting information may be found online in the Supporting Information section.

How to cite this article: Rocchini D, Thouverai E, Marcantonio M, et al. rasterdiv— An Information Theory tailored R package for measuring ecosystem heterogeneity from space: To the origin and back. Methods Ecol Evol. 2021;12:1093– 1102. https://doi.org/10.1111/2041- 210X.13583

Appendix 11

Mathematical proof of the equivalence between2

Renyi entropy and Shannon’s H given α→ 13

rasterdiv - an Information Theory tailored4

R package for measuring ecosystem5

heterogeneity from space: to the origin and6

back7

8

December 12, 20209

Theorem (De l’Hopital). Let f, g : (a, b) 7→ R be two functions such that10

• limx→a f(x) = limx→a g(x) = 011

• f and g are derivable in (a, b) with g′(x) 6= 0 for every x ∈ (a, b)12

• the limit limx→af ′(x)g′(x)

= L with L ∈ R13

then14

limx→a

f(x)

g(x)= L.

We now verify if the hypothesis hold in our case. Let f : (1,+∞) 7→ R15

and g : (1,+∞) 7→ R be functions such that f(x) = log(∑N

i=1 pxi ) and g(x) =16

1 − x for every x ∈ (1,+∞). Since pi is a fixed number in (0, 1) for every17

i ∈ {1, . . . , } with∑N

i=1 pi = 1, then18

f(1) = log(N∑i=1

pi) = 0.

Moreover, since f is a composition of derivable functions in (1,+∞), f19

is derivable in the definition domain. It is trivial that g(1) = 0, that g is20

1

derivable in the domain set and that g′(x) 6= 0 for every x ∈ (1,+∞). The21

limit of the derivatives ratio is:22

limx→1

f ′(x)

g′(x)= lim

x→1

(∑N

i=1 pxi )−1(

∑Ni=1 p

xi log pi)

1=

N∑i=1

pi log pi

which is finite due to the hypothesis over pi. Since all the hypotheses of De23

l’Hopital’s Theorem hold, the thesis follows, i.e.24

limα→1

Hα = H,

or Renyi ’s entropy equals to Shannon’s H for α→ 1.25

2